Abstract

We develop a coupled-wave model that is capable of treating finite-size square-lattice photonic crystal surface emitting lasers with transverse-electric polarization. Various properties of interest including threshold gain, mode frequency, field intensity envelope within the device, far-field pattern, as well as polarization and divergence angle of the output beam for the band-edge modes are calculated. Theoretical predictions of the lowest threshold mode and the output beam profile are in good agreement with our experimental findings. In particular, we show that, contrary to the infinite periodic case, the finite length of the device significantly affects surface emission and mode selection properties of the laser device.

© 2012 OSA

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  1. M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett.75, 316–318 (1999).
    [CrossRef]
  2. S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
    [CrossRef] [PubMed]
  3. M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B65, 195306 (2002).
    [CrossRef]
  4. I. Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron.39, 689–700 (2003).
    [CrossRef]
  5. D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express12, 1562–1568 (2004).
    [CrossRef] [PubMed]
  6. E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature441, 946 (2006).
    [CrossRef] [PubMed]
  7. M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett.88, 191105 (2006).
    [CrossRef]
  8. G. Xu, Y. Chassagneux, R. Colombelli, G. Beaudoin, and I. Sagnes, “Polarized single-lobed surface emission in mid-infrared, photonic-crystal, quantum-cascade lasers,” Opt. Lett.35, 859 (2010).
    [CrossRef] [PubMed]
  9. L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, and J. Faist, “Terahertz quantum cascade lasers based on two-dimensional photonic crystal resonators,” Opt. Express16, 5206–5217 (2008).
    [CrossRef] [PubMed]
  10. Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature457, 174–178 (2009).
    [CrossRef] [PubMed]
  11. L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photon. Rev.5, 647–658 (2011).
  12. H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science319, 445–447 (2008).
    [CrossRef]
  13. Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics4, 447–450 (2010).
    [CrossRef]
  14. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B44, 8565–8571 (1991).
    [CrossRef]
  15. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B65, 235112 (2002).
    [CrossRef]
  16. H. Y. Ryu, M. Notomi, and Y. H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B68, 045209 (2003).
    [CrossRef]
  17. M. Yokoyama and S. Noda, “Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser,” Opt. Express13, 2869–2880 (2005).
    [CrossRef] [PubMed]
  18. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys.43, 2327–2335 (1972).
    [CrossRef]
  19. M. Toda, “Proposed cross grating single-mode DFB laser,” IEEE J. Quantum Electron.28, 1653–1662 (1992).
    [CrossRef]
  20. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett.89, 021101 (2006).
    [CrossRef]
  21. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron.46, 788–795 (2010).
    [CrossRef]
  22. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: a general approach,” Phys. Rev. B84, 195119 (2011).
    [CrossRef]
  23. K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun.23, 1335–1340 (2005).
    [CrossRef]
  24. W. Kunishi, D. Ohnishi, E. Miyai, K. Sakai, and S. Noda, “High-power single-lobed surface-emitting photonic-crystal laser,” Conference on Lasers and Electro-Optics (CLEO), CMKK1, Long Beach, May, 2006.
  25. W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron.13, 134–141 (1977).
    [CrossRef]
  26. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air-holes with arbitrary sidewalls,” Opt. Express19, 24672–24686 (2011).
    [CrossRef] [PubMed]
  27. M. J. Bergmann and H. C. Casey, “Optical-field calculations for lossy multiple-layer AlGl-1N/InxG1-xN laser diodes,” J. Appl. Phys.84, 1196–1203 (1998).
    [CrossRef]
  28. D. H. Sheen, K. Tuncay, C. E. Baag, and P. J. Ortoleva, “Parallel implemantation of a velocity-stress staggered-grid finite-difference method for 2-D poroelastic wave propagation,” Comput. Geosci.32, 1182–1191 (2006).
    [CrossRef]
  29. E. Miyai and S. Noda, “Phase-shift effect on a two-dimensional surface-emitting photonic-crystal laser,” Appl. Phys. Lett.86, 111113 (2005).
    [CrossRef]
  30. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, 2007).
  31. Fundamentally, radiation fields emitted from the center of the laser cavity have similar properties to those emitted from an infinite periodic structure described in Ref. [22]. Unlike CC air holes, Fourier coefficients (ξm,n) of the dielectric function ε(r) for ET air holes are complex numbers. Therefore, the radiation field intensity is proportional to |ξ−1,0Rx + ξ1,0Sx|2 [see Eq. (A4) in Appendix A and Eq. (B11) in Appendix B]. These complex Fourier coefficient terms multiplied to basic waves may change the phase difference of the waves diffracted vertically, resulting in a suppression of the destructive interference.
  32. H. A. Haus, “Gain saturation in distributed feedback lasers,” Appl. Opt.14, 2650–2652 (1975).
    [CrossRef] [PubMed]
  33. S. H. Macomber, “Nonlinear analysis of surface-emitting distributed feedback lasers,” IEEE J. Quantum Electron.26, 2065–2074 (1990).
    [CrossRef]

2011

L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photon. Rev.5, 647–658 (2011).

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: a general approach,” Phys. Rev. B84, 195119 (2011).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air-holes with arbitrary sidewalls,” Opt. Express19, 24672–24686 (2011).
[CrossRef] [PubMed]

2010

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron.46, 788–795 (2010).
[CrossRef]

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics4, 447–450 (2010).
[CrossRef]

G. Xu, Y. Chassagneux, R. Colombelli, G. Beaudoin, and I. Sagnes, “Polarized single-lobed surface emission in mid-infrared, photonic-crystal, quantum-cascade lasers,” Opt. Lett.35, 859 (2010).
[CrossRef] [PubMed]

2009

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature457, 174–178 (2009).
[CrossRef] [PubMed]

2008

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science319, 445–447 (2008).
[CrossRef]

L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, and J. Faist, “Terahertz quantum cascade lasers based on two-dimensional photonic crystal resonators,” Opt. Express16, 5206–5217 (2008).
[CrossRef] [PubMed]

2006

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature441, 946 (2006).
[CrossRef] [PubMed]

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett.88, 191105 (2006).
[CrossRef]

D. H. Sheen, K. Tuncay, C. E. Baag, and P. J. Ortoleva, “Parallel implemantation of a velocity-stress staggered-grid finite-difference method for 2-D poroelastic wave propagation,” Comput. Geosci.32, 1182–1191 (2006).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett.89, 021101 (2006).
[CrossRef]

2005

M. Yokoyama and S. Noda, “Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser,” Opt. Express13, 2869–2880 (2005).
[CrossRef] [PubMed]

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun.23, 1335–1340 (2005).
[CrossRef]

E. Miyai and S. Noda, “Phase-shift effect on a two-dimensional surface-emitting photonic-crystal laser,” Appl. Phys. Lett.86, 111113 (2005).
[CrossRef]

2004

2003

H. Y. Ryu, M. Notomi, and Y. H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B68, 045209 (2003).
[CrossRef]

I. Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron.39, 689–700 (2003).
[CrossRef]

2002

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B65, 195306 (2002).
[CrossRef]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B65, 235112 (2002).
[CrossRef]

2001

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
[CrossRef] [PubMed]

1999

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett.75, 316–318 (1999).
[CrossRef]

1998

M. J. Bergmann and H. C. Casey, “Optical-field calculations for lossy multiple-layer AlGl-1N/InxG1-xN laser diodes,” J. Appl. Phys.84, 1196–1203 (1998).
[CrossRef]

1992

M. Toda, “Proposed cross grating single-mode DFB laser,” IEEE J. Quantum Electron.28, 1653–1662 (1992).
[CrossRef]

1991

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B44, 8565–8571 (1991).
[CrossRef]

1990

S. H. Macomber, “Nonlinear analysis of surface-emitting distributed feedback lasers,” IEEE J. Quantum Electron.26, 2065–2074 (1990).
[CrossRef]

1977

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron.13, 134–141 (1977).
[CrossRef]

1975

1972

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys.43, 2327–2335 (1972).
[CrossRef]

Amanti, M. I.

Baag, C. E.

D. H. Sheen, K. Tuncay, C. E. Baag, and P. J. Ortoleva, “Parallel implemantation of a velocity-stress staggered-grid finite-difference method for 2-D poroelastic wave propagation,” Comput. Geosci.32, 1182–1191 (2006).
[CrossRef]

Barbieri, S.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature457, 174–178 (2009).
[CrossRef] [PubMed]

Beaudoin, G.

Beere, H. E.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature457, 174–178 (2009).
[CrossRef] [PubMed]

Bergmann, M. J.

M. J. Bergmann and H. C. Casey, “Optical-field calculations for lossy multiple-layer AlGl-1N/InxG1-xN laser diodes,” J. Appl. Phys.84, 1196–1203 (1998).
[CrossRef]

Bewley, W. W.

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett.88, 191105 (2006).
[CrossRef]

Burnham, R. D.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron.13, 134–141 (1977).
[CrossRef]

Canedy, C. L.

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett.88, 191105 (2006).
[CrossRef]

Casey, H. C.

M. J. Bergmann and H. C. Casey, “Optical-field calculations for lossy multiple-layer AlGl-1N/InxG1-xN laser diodes,” J. Appl. Phys.84, 1196–1203 (1998).
[CrossRef]

Chassagneux, Y.

G. Xu, Y. Chassagneux, R. Colombelli, G. Beaudoin, and I. Sagnes, “Polarized single-lobed surface emission in mid-infrared, photonic-crystal, quantum-cascade lasers,” Opt. Lett.35, 859 (2010).
[CrossRef] [PubMed]

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature457, 174–178 (2009).
[CrossRef] [PubMed]

Chutinan, A.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B65, 195306 (2002).
[CrossRef]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
[CrossRef] [PubMed]

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett.75, 316–318 (1999).
[CrossRef]

Colombelli, R.

G. Xu, Y. Chassagneux, R. Colombelli, G. Beaudoin, and I. Sagnes, “Polarized single-lobed surface emission in mid-infrared, photonic-crystal, quantum-cascade lasers,” Opt. Lett.35, 859 (2010).
[CrossRef] [PubMed]

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature457, 174–178 (2009).
[CrossRef] [PubMed]

Davies, A. G.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature457, 174–178 (2009).
[CrossRef] [PubMed]

Faist, J.

Fan, S.

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B65, 235112 (2002).
[CrossRef]

Giovannini, M.

Haus, H. A.

Imada, M.

D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express12, 1562–1568 (2004).
[CrossRef] [PubMed]

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B65, 195306 (2002).
[CrossRef]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
[CrossRef] [PubMed]

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett.75, 316–318 (1999).
[CrossRef]

Iwahashi, S.

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air-holes with arbitrary sidewalls,” Opt. Express19, 24672–24686 (2011).
[CrossRef] [PubMed]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: a general approach,” Phys. Rev. B84, 195119 (2011).
[CrossRef]

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics4, 447–450 (2010).
[CrossRef]

Jianglin, Y.

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science319, 445–447 (2008).
[CrossRef]

Joannopoulos, J. D.

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B65, 235112 (2002).
[CrossRef]

Khanna, S. P.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature457, 174–178 (2009).
[CrossRef] [PubMed]

Kim, C. S.

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett.88, 191105 (2006).
[CrossRef]

Kim, M.

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett.88, 191105 (2006).
[CrossRef]

Kogelnik, H.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys.43, 2327–2335 (1972).
[CrossRef]

Kunishi, W.

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics4, 447–450 (2010).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature441, 946 (2006).
[CrossRef] [PubMed]

W. Kunishi, D. Ohnishi, E. Miyai, K. Sakai, and S. Noda, “High-power single-lobed surface-emitting photonic-crystal laser,” Conference on Lasers and Electro-Optics (CLEO), CMKK1, Long Beach, May, 2006.

Kurosaka, Y.

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics4, 447–450 (2010).
[CrossRef]

Lee, Y. H.

H. Y. Ryu, M. Notomi, and Y. H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B68, 045209 (2003).
[CrossRef]

Liang, Y.

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: a general approach,” Phys. Rev. B84, 195119 (2011).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air-holes with arbitrary sidewalls,” Opt. Express19, 24672–24686 (2011).
[CrossRef] [PubMed]

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics4, 447–450 (2010).
[CrossRef]

Lindle, J. R.

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett.88, 191105 (2006).
[CrossRef]

Linfield, E. H.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature457, 174–178 (2009).
[CrossRef] [PubMed]

Macomber, S. H.

S. H. Macomber, “Nonlinear analysis of surface-emitting distributed feedback lasers,” IEEE J. Quantum Electron.26, 2065–2074 (1990).
[CrossRef]

Mahler, L.

L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photon. Rev.5, 647–658 (2011).

Maineult, W.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature457, 174–178 (2009).
[CrossRef] [PubMed]

Maradudin, A. A.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B44, 8565–8571 (1991).
[CrossRef]

Matsubara, H.

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science319, 445–447 (2008).
[CrossRef]

Meyer, J. R.

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett.88, 191105 (2006).
[CrossRef]

I. Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron.39, 689–700 (2003).
[CrossRef]

Miyai, E.

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics4, 447–450 (2010).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron.46, 788–795 (2010).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett.89, 021101 (2006).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature441, 946 (2006).
[CrossRef] [PubMed]

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun.23, 1335–1340 (2005).
[CrossRef]

E. Miyai and S. Noda, “Phase-shift effect on a two-dimensional surface-emitting photonic-crystal laser,” Appl. Phys. Lett.86, 111113 (2005).
[CrossRef]

W. Kunishi, D. Ohnishi, E. Miyai, K. Sakai, and S. Noda, “High-power single-lobed surface-emitting photonic-crystal laser,” Conference on Lasers and Electro-Optics (CLEO), CMKK1, Long Beach, May, 2006.

Mochizuki, M.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B65, 195306 (2002).
[CrossRef]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
[CrossRef] [PubMed]

Murata, M.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett.75, 316–318 (1999).
[CrossRef]

Noda, S.

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: a general approach,” Phys. Rev. B84, 195119 (2011).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air-holes with arbitrary sidewalls,” Opt. Express19, 24672–24686 (2011).
[CrossRef] [PubMed]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron.46, 788–795 (2010).
[CrossRef]

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics4, 447–450 (2010).
[CrossRef]

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science319, 445–447 (2008).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature441, 946 (2006).
[CrossRef] [PubMed]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett.89, 021101 (2006).
[CrossRef]

M. Yokoyama and S. Noda, “Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser,” Opt. Express13, 2869–2880 (2005).
[CrossRef] [PubMed]

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun.23, 1335–1340 (2005).
[CrossRef]

E. Miyai and S. Noda, “Phase-shift effect on a two-dimensional surface-emitting photonic-crystal laser,” Appl. Phys. Lett.86, 111113 (2005).
[CrossRef]

D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express12, 1562–1568 (2004).
[CrossRef] [PubMed]

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B65, 195306 (2002).
[CrossRef]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
[CrossRef] [PubMed]

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett.75, 316–318 (1999).
[CrossRef]

W. Kunishi, D. Ohnishi, E. Miyai, K. Sakai, and S. Noda, “High-power single-lobed surface-emitting photonic-crystal laser,” Conference on Lasers and Electro-Optics (CLEO), CMKK1, Long Beach, May, 2006.

Notomi, M.

H. Y. Ryu, M. Notomi, and Y. H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B68, 045209 (2003).
[CrossRef]

Ohnishi, D.

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics4, 447–450 (2010).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature441, 946 (2006).
[CrossRef] [PubMed]

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun.23, 1335–1340 (2005).
[CrossRef]

D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express12, 1562–1568 (2004).
[CrossRef] [PubMed]

W. Kunishi, D. Ohnishi, E. Miyai, K. Sakai, and S. Noda, “High-power single-lobed surface-emitting photonic-crystal laser,” Conference on Lasers and Electro-Optics (CLEO), CMKK1, Long Beach, May, 2006.

Okano, T.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature441, 946 (2006).
[CrossRef] [PubMed]

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun.23, 1335–1340 (2005).
[CrossRef]

D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express12, 1562–1568 (2004).
[CrossRef] [PubMed]

Ortoleva, P. J.

D. H. Sheen, K. Tuncay, C. E. Baag, and P. J. Ortoleva, “Parallel implemantation of a velocity-stress staggered-grid finite-difference method for 2-D poroelastic wave propagation,” Comput. Geosci.32, 1182–1191 (2006).
[CrossRef]

Peng, C.

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air-holes with arbitrary sidewalls,” Opt. Express19, 24672–24686 (2011).
[CrossRef] [PubMed]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: a general approach,” Phys. Rev. B84, 195119 (2011).
[CrossRef]

Plihal, M.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B44, 8565–8571 (1991).
[CrossRef]

Ritchie, D. A.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature457, 174–178 (2009).
[CrossRef] [PubMed]

Ryu, H. Y.

H. Y. Ryu, M. Notomi, and Y. H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B68, 045209 (2003).
[CrossRef]

Sagnes, I.

Saito, H.

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science319, 445–447 (2008).
[CrossRef]

Sakaguchi, T.

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun.23, 1335–1340 (2005).
[CrossRef]

Sakai, K.

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: a general approach,” Phys. Rev. B84, 195119 (2011).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air-holes with arbitrary sidewalls,” Opt. Express19, 24672–24686 (2011).
[CrossRef] [PubMed]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron.46, 788–795 (2010).
[CrossRef]

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics4, 447–450 (2010).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature441, 946 (2006).
[CrossRef] [PubMed]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett.89, 021101 (2006).
[CrossRef]

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun.23, 1335–1340 (2005).
[CrossRef]

W. Kunishi, D. Ohnishi, E. Miyai, K. Sakai, and S. Noda, “High-power single-lobed surface-emitting photonic-crystal laser,” Conference on Lasers and Electro-Optics (CLEO), CMKK1, Long Beach, May, 2006.

Sasaki, G.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett.75, 316–318 (1999).
[CrossRef]

Scifres, D. R.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron.13, 134–141 (1977).
[CrossRef]

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys.43, 2327–2335 (1972).
[CrossRef]

Sheen, D. H.

D. H. Sheen, K. Tuncay, C. E. Baag, and P. J. Ortoleva, “Parallel implemantation of a velocity-stress staggered-grid finite-difference method for 2-D poroelastic wave propagation,” Comput. Geosci.32, 1182–1191 (2006).
[CrossRef]

Sirigu, L.

Streifer, W.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron.13, 134–141 (1977).
[CrossRef]

Tanaka, Y.

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science319, 445–447 (2008).
[CrossRef]

Terazzi, R.

Toda, M.

M. Toda, “Proposed cross grating single-mode DFB laser,” IEEE J. Quantum Electron.28, 1653–1662 (1992).
[CrossRef]

Tokuda, T.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett.75, 316–318 (1999).
[CrossRef]

Tredicucci, A.

L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photon. Rev.5, 647–658 (2011).

Tuncay, K.

D. H. Sheen, K. Tuncay, C. E. Baag, and P. J. Ortoleva, “Parallel implemantation of a velocity-stress staggered-grid finite-difference method for 2-D poroelastic wave propagation,” Comput. Geosci.32, 1182–1191 (2006).
[CrossRef]

Vurgaftman, I.

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett.88, 191105 (2006).
[CrossRef]

I. Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron.39, 689–700 (2003).
[CrossRef]

Xu, G.

Yariv, A.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, 2007).

Yeh, P.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, 2007).

Yokoyama, M.

M. Yokoyama and S. Noda, “Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser,” Opt. Express13, 2869–2880 (2005).
[CrossRef] [PubMed]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
[CrossRef] [PubMed]

Yoshimoto, S.

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science319, 445–447 (2008).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

E. Miyai and S. Noda, “Phase-shift effect on a two-dimensional surface-emitting photonic-crystal laser,” Appl. Phys. Lett.86, 111113 (2005).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett.89, 021101 (2006).
[CrossRef]

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett.75, 316–318 (1999).
[CrossRef]

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett.88, 191105 (2006).
[CrossRef]

Comput. Geosci.

D. H. Sheen, K. Tuncay, C. E. Baag, and P. J. Ortoleva, “Parallel implemantation of a velocity-stress staggered-grid finite-difference method for 2-D poroelastic wave propagation,” Comput. Geosci.32, 1182–1191 (2006).
[CrossRef]

IEEE J. Quantum Electron.

S. H. Macomber, “Nonlinear analysis of surface-emitting distributed feedback lasers,” IEEE J. Quantum Electron.26, 2065–2074 (1990).
[CrossRef]

M. Toda, “Proposed cross grating single-mode DFB laser,” IEEE J. Quantum Electron.28, 1653–1662 (1992).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron.46, 788–795 (2010).
[CrossRef]

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron.13, 134–141 (1977).
[CrossRef]

I. Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron.39, 689–700 (2003).
[CrossRef]

IEEE J. Sel. Areas Commun.

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun.23, 1335–1340 (2005).
[CrossRef]

J. Appl. Phys.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys.43, 2327–2335 (1972).
[CrossRef]

M. J. Bergmann and H. C. Casey, “Optical-field calculations for lossy multiple-layer AlGl-1N/InxG1-xN laser diodes,” J. Appl. Phys.84, 1196–1203 (1998).
[CrossRef]

Laser Photon. Rev.

L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photon. Rev.5, 647–658 (2011).

Nat. Photonics

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics4, 447–450 (2010).
[CrossRef]

Nature

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature457, 174–178 (2009).
[CrossRef] [PubMed]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature441, 946 (2006).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. B

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B65, 195306 (2002).
[CrossRef]

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B44, 8565–8571 (1991).
[CrossRef]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B65, 235112 (2002).
[CrossRef]

H. Y. Ryu, M. Notomi, and Y. H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B68, 045209 (2003).
[CrossRef]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: a general approach,” Phys. Rev. B84, 195119 (2011).
[CrossRef]

Science

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science319, 445–447 (2008).
[CrossRef]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
[CrossRef] [PubMed]

Other

W. Kunishi, D. Ohnishi, E. Miyai, K. Sakai, and S. Noda, “High-power single-lobed surface-emitting photonic-crystal laser,” Conference on Lasers and Electro-Optics (CLEO), CMKK1, Long Beach, May, 2006.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, 2007).

Fundamentally, radiation fields emitted from the center of the laser cavity have similar properties to those emitted from an infinite periodic structure described in Ref. [22]. Unlike CC air holes, Fourier coefficients (ξm,n) of the dielectric function ε(r) for ET air holes are complex numbers. Therefore, the radiation field intensity is proportional to |ξ−1,0Rx + ξ1,0Sx|2 [see Eq. (A4) in Appendix A and Eq. (B11) in Appendix B]. These complex Fourier coefficient terms multiplied to basic waves may change the phase difference of the waves diffracted vertically, resulting in a suppression of the destructive interference.

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic structure of square-lattice PC-SEL device with circular (CC) and equilateral triangular (ET) air holes (inset: scanning electron microscope images). (b) A typical photonic-band structure calculated by 2D-PWEM. There exist four band-edge modes in the vicinity of the second-order Γ point, which we refer to as modes A, B, C and D, in order of increasing frequency.

Fig. 2
Fig. 2

(i) Normalized threshold gain (αL) as a function of normalized mode frequency deviation (δL) for a PC-SEL with ET air holes. The four band-edge modes A–D are indicated by arrows. (ii) Field intensity envelopes of the individual band-edge modes. (iii) Field distributions inside a unit cell located at the center of the cavity, in which colors and arrows represent H- and E-fields, respectively. Thick black triangles indicate the air holes. In the calculations, we use the structural parameters shown in Tab. 1, the air-hole filling factor f = 0.16, the lattice constant a=295 nm, and the device length L=70 μm.The laser is divided into 14 sections for which the eigenvalues (αL and δL) converge well.

Fig. 3
Fig. 3

Calculated far-field patterns (FFPs) of mode A (i.e. the lowest threshold mode) for (a) CC and (b) ET air holes and experimentally observed FFPs for (c) CC and (d) ET air holes. The insets in (a) and (b) represent the x and y components of the far field. Parameters used for the calculations are the same as those shown in Fig. 2. The yellow arrows in (c) and (d) indicate the directions of polarization.

Fig. 4
Fig. 4

(a) Illustrative field distribution patterns (plotted in a single unit cell) of mode A for CC air holes at the center of the cavity (i): ( L 2, L 2) and toward the edges (ii): ( 6 L 7, L 2), (iii): ( L 7, L 2, (iv): ( L 2, 6 L 7), and (vi): ( L 2, L 7). Colors and arrows represent H- and E-fields, respectively. Thick black circles indicate the locations of air holes with respect to the unit cell. (b) Field intensities of basic waves propagating in the ±x directions, |Rx|2 and |Sx|2 (red and blue curves), and the radiation field intensity represented by |Rx + Sx|2 (black curve) along the axis y = L/2. Note that Rx and Sx are zero at x = 0 and x = L, respectively, which corresponds to the boundary condition described by Eq. (22).

Fig. 5
Fig. 5

Normalized mode frequency (a, b) and threshold gain (c, d) as a function of the device length L for CC and ET air holes (f = 0.16 and a=295 nm). The solid curves and dashed lines are calculated for finite and infinite structures, respectively.

Tables (3)

Tables Icon

Table 1 Structural Parameters of the 2D PC-SEL Device

Tables Icon

Table 2 Normalized Threshold Gain (αL) of the Four Band-Edge Modes A–D for CC and ET Air-Hole Shapes (f = 0.16, a=295 nm, and L=70 μm)

Tables Icon

Table 3 Percentage of the In-Plane Loss with Respect to the Total Stimulated Power, Pedge/Pstim for CC and ET Air Holes with Different Device Lengths (f = 0.16 and a=295 nm)

Equations (55)

Equations on this page are rendered with MathJax. Learn more.

× × E ( r ) = k 0 2 n ˜ 2 ( r ) E ( r ) ,
E j ( r ) = m , n E j , m , n ( x , y , z ) e im β 0 x in β 0 y , j = x , y ,
n 2 ( r ) = n 0 2 ( z ) + m 0 , n 0 ξ m , n ( z ) e im β 0 x in β 0 y .
[ 2 z 2 + k 0 2 n 0 2 ( z ) + 2 i k 0 n 0 ( z ) α ˜ ( z ) n 2 β 0 2 ] E x , m , n 2 in β 0 E x , m , n y + 2 E x , m , n y 2 + m n β 0 2 E y , m , n 2 E y , m , n x y + i β 0 ( m E y , m , n y + n E y , m , n x ) = k 0 2 m m , n n ξ m m , n n E x , m , n ,
[ 2 z 2 + k 0 2 n 0 2 ( z ) + 2 i k 0 n 0 ( z ) α ˜ ( z ) m 2 β 0 2 ] E y , m , n 2 im β 0 E y , m , n x + 2 E y , m , n x 2 + m n β 0 2 E x , m , n 2 E x , m , n x y + i β 0 ( m E x , m , n y + n E x , m , n x ) = k 0 2 m m , n n ξ m m , n n E y , m , n ,
z [ ( E x , m , n x + E y , m , n y ) i β 0 ( m E x , m n + n E y , m n ) ] = 0.
E x , 1 , 0 = 0 , E y , 1 , 0 = R x ( x , y ) Θ 0 ( z ) ,
E x , 1 , 0 = 0 , E y , 1 , 0 = S x ( x , y ) Θ 0 ( z ) ,
E x , 0 , 1 = R y ( x , y ) Θ 0 ( z ) , E y , 0 , 1 = 0.
E x , 0 , 1 = S y ( x , y ) Θ 0 ( z ) , E y , 0 , 1 = 0 .
2 Θ 0 z 2 + [ k 0 2 n 0 2 ( z ) β 2 ] Θ 0 = 0 ,
[ 2 Θ 0 z 2 + ( k 0 2 n 0 2 ( z ) + 2 i k 0 n 0 ( z ) α ˜ ( z ) β 0 2 ) Θ 0 ] R x 2 i β 0 R x x Θ 0 = k 0 2 m 1 , n 0 ξ 1 m , n E y , m , n .
( β 2 β 0 2 ) R x Θ 0 + 2 i k 0 n 0 ( z ) α ˜ ( z ) R x Θ 0 2 i β 0 R x x Θ 0 = k 0 2 m 1 , n 0 ξ 1 m , n E y , m , n .
E x , 0 , 0 = Δ E x ( z ) , E y , 0 , 0 = Δ E y ( z ) .
i R x x + ( δ + i α ) R x = κ 2 , 0 S x k 0 2 2 β 0 ξ 1 , 0 P C Δ E y ( z ) Θ 0 * ( z ) d z k 0 2 2 β 0 m 2 + n 2 > 1 ξ 1 m , n P C E y , m , n ( z ) Θ 0 * ( z ) d z ,
i S x x + ( δ + i α ) S x = κ 2 , 0 R x k 0 2 2 β 0 ξ 1 , 0 P C Δ E y ( z ) Θ 0 * ( z ) d z k 0 2 2 β 0 m 2 + n 2 > 1 ξ 1 m , n P C E y , m , n ( z ) Θ 0 * ( z ) d z ,
i R y y + ( δ + i α ) R y = κ 0 , 2 S y k 0 2 2 β 0 ξ 0 , 1 P C Δ E x ( z ) Θ 0 * ( z ) d z k 0 2 2 β 0 m 2 + n 2 > 1 ξ m , 1 n P C E x , m , n ( z ) Θ 0 * ( z ) d z ,
i S y y + ( δ + i α ) S y = κ 0 , 2 R y k 0 2 2 β 0 ξ 0 , 1 P C Δ E x ( z ) Θ 0 * ( z ) d z k 0 2 2 β 0 m 2 + n 2 > 1 ξ m , 1 n P C E x , m , n ( z ) Θ 0 * ( z ) d z .
κ ± 2 , 0 = k 0 2 2 β 0 ξ ± 2 , 0 P C | Θ 0 ( z ) | 2 d z ,
κ 0 , ± 2 = k 0 2 2 β 0 ξ 0 , ± 2 P C | Θ 0 ( z ) | 2 d z ,
( δ + i α ) ( R x S x R y S y ) = C ( R x S x R y S y ) + i ( R x / x S x / x R y / y S y / y ) .
R x ( 0 , y ) = S x ( L , y ) = R y ( x , 0 ) = S y ( x , L ) = 0 ,
I ( x , y ) = | R x ( x , y ) | 2 + | S x ( x , y ) | 2 + | R y ( x , y ) | 2 + | S y ( x , y ) | 2 .
F j ( θ x , θ y , t ) ( cos θ x + cos θ y 1 ) 0 L Δ E j ( x , y , z = d P C / 2 ) e i ω t e i k 0 ( tan θ x x + tan θ y y ) ] d x d y , j = x , y ,
B ( θ x , θ y ) = B x ( θ x , θ y ) + B y ( θ x , θ y ) ,
B j ( θ x , θ y ) = 1 T 0 T | Re [ F j ( θ x , θ y , t ) ] | 2 d t , j = x , y , T = 2 π / ω .
k 0 sin ( δ θ ) = δ k ,
P stim = P edge + P rad .
P edge / P stim = 1 P rad / P stim .
[ 2 z 2 + k 0 2 n 0 2 ( z ) n 2 β 0 2 ] E x , m , n + m n β 0 2 E y , m , n = k 0 2 m m , n n ξ m m , n n E x , m , n ,
[ 2 z 2 + k 0 2 n 0 2 ( z ) m 2 β 0 2 ] E y , m , n + m n β 0 2 E x , m , n = k 0 2 m m , n n ξ m m , n n E y , m , n ,
z [ m E x , m n + n E y , m n ] = 0 ,
Δ E x ( x , y , z ) = [ ξ 0 , 1 R y ( x , y ) + ξ 0 , 1 S y ( x , y ) ] k 0 2 P C G ( z , z ) Θ 0 ( z ) d z , Δ E y ( x , y , z ) = [ ξ 1 , 0 R x ( x , y ) + ξ 1 , 0 S x ( x , y ) ] k 0 2 P C G ( z , z ) Θ 0 ( z ) d z .
( P C E x , m , n ( z ) Θ 0 * ( z ) d z P C E y , m , n ( z ) Θ 0 * ( z ) d z ) = 1 m 2 + n 2 ( n m m n ) ( m μ m , n ( 1 , 0 ) m μ m , n ( 1 , 0 ) n μ m , n ( 0 , 1 ) n μ m , n ( 0 , 1 ) n ν m , n ( 1 , 0 ) n ν m , n ( 1 , 0 ) m ν m , n ( 0 , 1 ) m ν m , n ( 0 , 1 ) ) V , ( ς x , m , n ( 1 , 0 ) ς x , m , n ( 1 , 0 ) ς x , m , n ( 0 , 1 ) ς x , m , n ( 0 , 1 ) ς y , m , n ( 1 , 0 ) ς y , m , n ( 1 , 0 ) ς y , m , n ( 0 , 1 ) ς y , m , n ( 0 , 1 ) ) V ,
V = ( R x , S x , R y , S y ) t ,
μ m , n ( r , s ) = k 0 2 P C ξ m r , n s ( z ) G m , n ( z , z ) Θ 0 ( z ) Θ 0 * ( z ) d z d z ,
ν m , n ( r , s ) = P C 1 n 0 2 ( z ) ξ m r , n s ( z ) | Θ 0 ( z ) | 2 d z ,
G m , n ( z , z ) = 1 2 β z , m , n e β z , m , n | z z | , β z,m,n = ( m 2 + n 2 ) β 0 2 k 0 2 n 0 2 ( z ) .
C = C 1 D + C rad + C 2 D ,
C 1 D = ( 0 κ 2 , 0 0 0 κ 2 , 0 0 0 0 0 0 0 κ 0 , 2 0 0 κ 0 , 2 0 ) ,
C rad = ( ζ 1 , 0 ( 1 , 0 ) ζ 1 , 0 ( 1 , 0 ) 0 0 ζ 1 , 0 ( 1 , 0 ) ζ 1 , 0 ( 1 , 0 ) 0 0 0 0 ζ 0 , 1 ( 0 , 1 ) ζ 0 , 1 ( 0 , 1 ) 0 0 ζ 0 , 1 ( 0 , 1 ) ζ 0 , 1 ( 0 , 1 ) ) ,
C 2 D = ( χ y , 1 , 0 ( 1 , 0 ) χ y , 1 , 0 ( 1 , 0 ) χ y , 1 , 0 ( 0 , 1 ) χ y , 1 , 0 ( 0 , 1 ) χ y , 1 , 0 ( 1 , 0 ) χ y , 1 , 0 ( 1 , 0 ) χ y , 1 , 0 ( 0 , 1 ) χ y , 1 , 0 ( 0 , 1 ) χ x , 0 , 1 ( 1 , 0 ) χ x , 0 , 1 ( 1 , 0 ) χ x , 0 , 1 ( 0 , 1 ) χ x , 0 , 1 ( 0 , 1 ) χ x , 0 , 1 ( 1 , 0 ) χ x , 0 , 1 ( 1 , 0 ) χ x , 0 , 1 ( 0 , 1 ) χ x , 0 , 1 ( 0 , 1 ) ) ,
ζ p , q ( r , s ) = k 0 4 2 β 0 P C ξ p , q ξ r , s G ( z , z ) Θ 0 ( z ) Θ 0 * ( z ) d z d z ,
χ j , p , q ( r , s ) = k 0 2 2 β 0 m 2 + n 2 > 1 ξ p m , q n ς j , m , n ( r , s ) , j = x , y .
i ( δ + i α ) R x = R x / x i ( C 11 R x + C 12 S x + C 13 R y + C 14 S y ) ,
i ( δ + i α ) S x = S x / x i ( C 21 R x + C 22 S x + C 23 R y + C 24 S y ) ,
i ( δ + i α ) R y = R y / y i ( C 31 R x + C 32 S x + C 33 R y + C 34 S y ) ,
i ( δ + i α ) S y = S y / y i ( C 41 R x + C 42 S x + C 43 R y + C 44 S y ) ,
2 α ( | R x | 2 + | S x | 2 + | R y | 2 + | S y | 2 ) = x ( | R x | 2 | S x | 2 ) + y ( | R y | 2 | S y | 2 ) + 2 κ v , i ( | ξ 1 , 0 R x + ξ 1 , 0 S x | 2 + | ξ 0 , 1 R y + ξ 0 , 1 S y | 2 ) ,
2 α 0 L ( | R x | 2 + | S x | 2 , | R y | 2 + | S y | 2 ) d x d y = 0 L [ x ( | R x | 2 | S x | 2 ) + y ( | R y | 2 | S y | 2 ) ] d x d y + 2 κ v , i ( | ξ 1 , 0 R x + ξ 1 , 0 S x | 2 + | ξ 0 , 1 R y + ξ 0 , 1 S y | 2 ) d x d y .
R x ( 0 , y ) = 0 , S x ( 0 , y ) = S ex ( y ) , R x ( L , y ) = R ex ( y ) , S x ( L , y ) = 0 , R y ( x , 0 ) = 0 , S y ( x , 0 ) = S ey ( x ) , R y ( x , L ) = R ey ( x ) , S y ( x , L ) = 0.
P stim = P edge + P rad ,
P stim = 2 α 0 L ( | R x | 2 + | S x | 2 + | R y | 2 + | S y | 2 ) d x d y ,
P edge = 0 L ( | R ex | 2 + | S ex | 2 ) d y + 0 L ( | R e y | 2 + | S ey | 2 ) d x ,
P rad = 2 κ v , i 0 L ( | ξ 1 , 0 R x + ξ 1 , 0 S x | 2 + | ξ 0 , 1 R y + ξ 0 , 1 S y | 2 ) d x d y .

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